Nonadiabatic theory of the superconducting state

The nonadiabatic theory of superconductivity has provided a useful tool to investigate the superconducting properties of narrow-band systems where Fermi energy EF is comparable with the phonon frequencies ?ph . Here we present the extention of nonadiabatic theory to the superconducting state (T?Tc) and we derive the generalized Eliashberg equations that include the first nonadiabatic diagrams arising from the breakdown of Migdal's theorem. We show that the opening of the superconducting gap modifies the momentum-frequency structure of the electron-phonon vertex function with respect to normal state. We study the effects of nonadia- batic terms on the superconducting gap ? and on the ratio 2?/Tc in order to relate possible strong-coupling phenomenology, signalized by 2?/Tc?3.53, and nonadiabatic effects.